Spectral Mappings and Network Transformations

The transmission line matrix method (TLM) has developed in many interesting directions that we have not been able to discuss in this thesis. Many different types of structures have been proposed, in particular those for which the dependent variables are not interleaved. Given that we have shown that certain DWNs can be derived from MDKCs, just as MDWD networks are, it would be interesting to know whether the various TLM structures can be arrived at in a similar way. A compact circuit representation would empower an algorithm designer enormously, and would almost certainly make a useful tool for designing new structures which are potentially more efficient and which may have better numerical properties.

We note, however, that for a given circuit representation of a system of PDEs, it is not at all obvious which spectral mappings should be applied in order to give rise to a useful structure; indeed, in the case of the DWN discussed in §4.10, the correct spectral mappings were arrived through a chance encounter with formulae buried in the dark basement of an old paper [61]. It should be possible to elucidate the link to a certain degree; does a particular network topology imply a particular integration rule or mapping? This is important, because for a given system of PDEs, there is not a single MDKC representation; any rules or transformations from classical network theory can be used to manipulate the MDKC into an infinite number of new topologies, each of which, upon discretization, gives rise to a distinct numerical method.